ABSTRACT
Likewise, one may construct functions m and n as limits of the sequences (mn)" and (nn)", respectively. Now we put
Pz(t,s):= c(t,s)z(t,s)
(13.33) +16 l(t,8,T)z(T,s)dT +1~m(t,,,,O')z(t,O')da +161~ n(t,s,T,U)Z(T,U)oodT
and show that P E £pCC) and P = P. First of all, by the construction of the functions c, I, m and n, and the obvious estimate
IP"x(t, a) - Px(t, a)1 ~ Ic,,(t, a) - c(t, a)llx(t, a)\
+ld.\m,,(t, a, 0') - met, s, O')llx(t, 0')1 dO'
+1" ld.\n,,(t, s, T, 0') - net, s, T, O')lIx(T, 0')1 dO'dT (x E G(D» we see that P E £p(G). Moreover, we have
liP" - PII.c(c) ~ sup [Ic,,(t,s) - c(t,s)1 (f,_)eD
+1" I',,(t, s, T) - let, s, T)I dT +ld. 1m,,(t, s, 0') - met, s, 0')\ dO' +1" ld. 1n,,(t,s,T,0') - n(t,s,T,O')\dO'dT] .
